Arbitrary Reference Wylie Breckenridge and Ofra Magidor (Penultimate draft of paper to appear in Philosophical Studies Please cite the final published version) Two fundamental rules of reasoning are Universal Generalisation (UG) and Existential Instantiation (EI). The former is the rule that allows us, given that we stipulate that ABC is an arbitrary triangle and prove that ABC has a property p, to conclude that all triangles have p. The latter is the rule that allows us, given that there exists a continuous function, to stipulate that f is an (arbitrary) continuous function. Reasoning according to these rules (or, as Kit Fine calls it, instantial reasoning ) is ubiquitous in both formal and informal contexts. Yet applications of these rules involve stipulations (even if only implicitly) such as Let ABC be an arbitrary triangle, Let n be an arbitrary number, or Let John be an arbitrary Frenchman. 1 And the semantics underlying such stipulations are far from clear. What, for example, does n refer to following the stipulation that n be an arbitrary number? In this paper, we argue that n refers to a number - an ordinary, particular number such as 58 or 2,345,043. Which one? We do not and cannot know, because the reference of n is fixed arbitrarily. 2 Underlying this proposal is a more general thesis: Arbitrary Reference (AR): It is possible to fix the reference of an expression arbitrarily. When we do so, the expression receives its ordinary kind of semantic-value, though we do not and cannot know which value in particular it receives. 3 Our aim in this paper is defend AR. In 1, we consider and respond to the most obvious objections to AR, clarifying the thesis in the process. The remaining two sections provide our positive case in favour of AR. In a nutshell, our positive argument is an inference to the best explanation: we argue that AR provides the best explanation of a range of phenomena. In 2 1 We do not need to always explicitly use the qualification arbitrary. Given the right context, we may simply use stipulations such as Let n be a natural number or Let Pierre be a Frenchman. We insert the explicit qualification simply to ensure one focuses on the relevant kinds of readings. 2 Knowledge which claims are notoriously context sensitive. When we say that we do not know which number n is, we mean that we cannot describe the number in some informative mode of presentation, such as n is 343. Of course we do know that n is n, that n refers to n, that n refers to whatever number it refers to, and so forth. This qualification should be kept in mind throughout the paper. 3 Except, of course, in special situations where we know there is only one value that the expression could receive (e.g. in the case of Let n be an arbitrary even prime number ). Page 1
we discuss the application of AR to instantial reasoning. We argue that AR can be used to develop a semantics which accounts for instantial reasoning, and moreover that this account is better than the prominent alternatives. In 3, we point to other applications that AR may have. The applications in this final section are more suggestive and require further development. Nevertheless, we think that they suffice to show that AR is a highly promising thesis, and that adopting it is likely to lay the path to a range of new solutions to some difficult philosophical puzzles. 1 Arbitrary Reference: objections and responses In this section we consider the most obvious objections to AR. If nothing else, the discussion should at least serve to clarify the thesis. Objection 1: AR cannot be correct because it violates some necessary conditions on reference. For an agent to refer to an object she must stand in some special kind of relation to the object: she must (in some sense) be acquainted with the object, or stand in some kind of casual relation to the object or (substitute your favourite condition on reference here). But often when one makes a stipulation such as Let Pierre be an arbitrary Frenchman, one does not stand in the appropriate relation to any particular Frenchman, and so one cannot succeed in referring to any particular Frenchman. Response 1: First, we are highly sceptical of any such necessary conditions on reference. For example, we maintain that the stipulation Let Julius refer to the actual tallest spy is entirely appropriate and, assuming that there is a unique tallest spy, we can successfully use it to fix the reference of Julius, even if we do not stand in any significant casual relation to that spy (or any other significant relation of acquaintance). 4 Second, even if there are such acquaintance-like conditions on reference, this would not really threaten AR. All we need to consider are examples in which one does stand in the relevant acquaintance-like relation to the object referred to. To take an extreme case, suppose that one maintains that in order to refer to a concrete object one needs to have seen that object. Now suppose that there are two bottles on the table in front of Alice and she can see each of them. Alice says to herself Let Jake be an arbitrary bottle on the table. According to 4 For a detailed defence of the claim that reference does not require such special necessary conditions see Hawthorne and Manley (MS). Page 2
AR, Jake (arbitrarily) refers to one of the bottles on the table. But Objection 1 does not apply to this case, because even the (artificially stringent) condition on reference envisaged is satisfied. The crucial point is this. It does not follow from our view that any stipulation of the form Let a be an arbitrary F successfully results in fixing a referent for a. One could plausibly require, for example, that if there are no Fs the stipulation fails and it is perfectly compatible with AR that it also fails when other, more complex conditions, do not obtain. All that AR requires is that at least some such stipulations succeed. Objection 2: If a stipulation such as Let Pierre be an arbitrary Frenchman succeeds in fixing the reference of Pierre to some particular Frenchman, then something must determine which Frenchman is referred to. But it is difficult to see what that would be. There doesn t seem to be anything about our behaviour, the sounds we utter, our brain states, or even our external environment that determines which Frenchman is referred to. Response 2: We agree that none of the factors mentioned determine which Frenchman is referred to. In fact, we propose that nothing determines which Frenchman is referred to nothing, that is, other than the semantic fact that we have referred to the particular Frenchman in question. 5 We simply deny that for it to be a fact that some particular Frenchman is being referred to, some other facts need to determine this fact. The mere claim that some facts are not determined by other facts is not in itself surprising. If the domain of facts is well-founded, then some facts are not such that they are true in virtue of other facts. Perhaps the worry is, though, that it there is something especially troubling about the claim that the particular fact in question (namely, who Pierre refers to) is not determined by other facts. We turn to this point in the next objection. 5 A tricky question is whether our intentions are sufficient to determine which Frenchman is being referred to. We certainly don t think that the reference is determined by any informative intentions of the sort I intend Pierre to refer to Jacques Chirac. But it may be that one has an intention that Pierre refers to Pierre or that Pierre refer to an arbitrary Frenchman, and that these in turn are sufficient to determine the reference of Pierre. But in so far as our intentions determine the reference in the latter way, there is still an important semantic fact (one concerning the word Pierre, or the phrase an arbitrary Frenchman, or analogous phrases in one s language of thought), a semantic fact that is crucial in determining the reference of Pierre and is not itself determined by non-semantic facts. Page 3
Objection 3: In response to Objection 2, you claim that when one stipulates that Pierre is an arbitrary Frenchman, nothing determines which Frenchman Pierre refers to. Let us concede the general point that some facts are not determined by other facts. But facts about reference are semantic facts, and it is standardly accepted that semantic facts are not primitive: rather they fully determined by use facts (broadly construed). In short, AR seems to contradict the platitude that semantic facts supervene on use facts. 6 Response 3: We accept that AR conflicts with the commonly-held view that semantic facts supervene on use facts. Indeed, one way to bring out this conflict is to consider possible worlds, w 1 and w 2, identical in all non-semantic respects, where in both Jill makes the stipulation Let Pierre be an arbitrary Frenchman. According to AR as we would like to think of it, it is possible that following Jill s stipulation, Pierre refers to one Frenchman (say Jacques Chirac) in w 1 but to another (say Nicolas Sarkozy) in w 2. Thus the two worlds are identical in their use facts (because they are identical in every non-semantic respect) but they differ in their semantic facts (because Pierre has a different referent in each of the two worlds). So if AR is correct then semantic facts do not supervene on use facts. We also recognise that Objection 3 is probably the most compelling objection to AR, and the main reason why it may be seen as radical. Nevertheless, we insist that the view that semantic facts supervene on use facts is simply incorrect. We rely here on the detailed defence of this claim that is provided in Kearns & Magidor (forthcoming). In particular, we note that this defence in no place relies on the truth of AR, and thus is not circular. 7 6 We allow that the objector interprets use facts in a broad enough way so as to include facts about one s environment or facts about which properties are most natural. It is clear, however, that the objector does not intend count as use facts such semantic or intentional facts as the fact that one uses Pierre to refer to, e.g., Jacques Chirac. For a more detailed discussion on how the claim that semantic facts supervene on use facts ought to be interpreted see Kearns & Magidor (forthcoming). 7 It is worth noting that one could also consider a different interpretation of AR, one according to which facts about arbitrary reference do supervene on use facts. One constraint on developing the theory in this manner is that to ensure that it will allow that not all stipulations of the form Let n be an arbitrary number result in n referring to the same number. (Otherwise one will have problems in applying AR to the case of instantial reasoning, in particular to stipulations such as Let n be an arbitrary number and let k be an arbitrary number). Two additional challenges for this interpretation of the view is to provide an explanation, on the one hand for how reference supervenes on use facts, and on the other hand for why despite the supervenience claim - we do not and cannot know the referent. One avenue to explore in this context is a brute supervenience view: one according to which the semantic facts supervene on use facts in an entirely unexplanatory manner and are hence unknowable (cf. Williamson (1994) and Cameron (2010)). But we find this position dialectically inferior to our own: if one is going to allow bruteness concerning the semantic realm into one s theory, why insist on the supervenience claim, rather than simply postulating brute contingent facts concerning reference, as our own theory does? After all, contingent brute facts seem less offensive than necessary ones. (See also Kearns & Page 4
Objection 4: You claim that following the above stipulation, Pierre refers to a particular Frenchman. How then, do you explain the fact that we do not and cannot know which Frenchman Pierre refers to? 8 Response 4: In a nutshell, our response is that we are ignorant of this fact precisely because nothing (non-semantic) determines which Frenchman Pierre refers to. Compare this with another case, one of determination over time. Suppose you are going to flip a coin (and assume that coin flips are genuinely non-deterministic). You do not and cannot know which side the coin will land. Why? The obvious answer is that this is because nothing in the current state of the world determines which side the coin will land. Of course, one could insist on a further explanation, namely an explanation of why you do not and cannot simply have direct knowledge of the primitive fact that coin will land on heads/tails. 9 Yet most of us seem content to simply shrug our shoulders at this stage: for some reason we simply don t have knowledge of such primitive facts about the future. Return to the case of arbitrary reference. You do not and cannot know what Pierre refers to. Why? The obvious answer is that this is because nothing in the state of the world determines what Pierre refers to. Of course, one could insist on a further explanation, namely an explanation of why you do not and cannot simply have direct knowledge of the primitive semantic fact that Pierre refers to so and so. Yet it seems that if we were happy to shrug our shoulders in the coin flipping case, we should be equally happy to shrug our shoulders in the case of arbitrary reference. One could of course point out to a disanalogy: in the case of the coin flip one could come to know the result of the flip after the coin-flip had happened, while in the case of arbitrary reference one cannot come to know which Frenchman Pierre refers to, even after the stipulation was made. This is correct, but misses the point of the original analogy. After the Magidor (forthcoming), 2.3 for a similar argument). At any rate, at least for the purposes of this paper, we have chosen to develop a version AR which denies supervenience. 8 It is worth point out in this context that our claim that one cannot know which Frenchman Pierre refers to is not intended to exclude scenarios where an omniscient being knows who Pierre refers to, or where one comes to know this fact by testimony of an omniscient being. When we say that we cannot know who Pierre refers to, we mean that we cannot know this in any reasonably ordinary scenario, one that does not involve omniscient beings. See Williamson (1997), p. 926 for a similar qualification regarding his epistemic view of vagueness. 9 We are assuming here that even in non-deterministic settings there are such facts about the future. Page 5
coin was flipped, there are facts about the world that determine that the coin was flipped in a certain way (e.g. the facts that the coin wasn t moved after the flip, and that it is now lying with its heads-side up). But even after the stipulation concerning Pierre was made, there are no facts (again no facts other than the fact that Pierre refers to whoever it refers to) that determine which Frenchman Pierre refers to, so it is no surprise that we cannot and do not know this fact. Objection 5: Even if the stipulation succeeds in fixing the reference of Pierre, it does not fix it to be some particular Frenchman. After all, the stipulation would have achieved exactly the same effect if instead of saying Let Pierre be an arbitrary Frenchman we would have said Let Pierre be an arbitrary Frenchman - no particular one. Response 5: We agree that the two stipulations would result in the reference of Pierre being fixed in exactly the same way, but we still maintain that it would have been fixed (even in the latter case) to a particular Frenchman. Pierre refers to a particular Frenchman. The qualification no particular one is a qualification of the manner of fixing the reference, rather than of the Frenchman referred to. That is to say, it points out that we are fixing the reference arbitrarily rather than particularly. This is not an uncommon use of particular in English. Consider for example the announcer of the results of a competition who says In no particular order, the winners are A, B, and C!. 10 In one sense, of course, the winners are given in a particular order: first A, then B, then C. But in another sense they are not: the order was chosen arbitrarily and not particularly. Similarly, we suggest, Pierre is a particular Frenchman in the former sense (there is some specific guy who is Pierre), but not in the latter sense (Pierre was picked arbitrarily and particularly). Objection 6: Even if the stipulation succeeds in fixing the reference of Pierre, it does not fix it to an ordinary particular Frenchman, but rather to a special kind of object: an arbitrary Frenchman. 11 Response 6: We are happy to concede that Pierre is an arbitrary Frenchman, but we deny that this means that Pierre is a special kind of object, one that is not an ordinary, flesh and blood, 10 We note that precisely this locution is used in the announcements of the results in the British television show The X factor. 11 Cf. the discussion of Fine s view in 2.1.3 below. Page 6
particular Frenchman. On our view an arbitrary Frenchman is just a Frenchman that is referred to arbitrarily. Compare this to the following cases. Suppose you make an intentional mistake. It is perfectly true that the mistake you made is intentional. But this does not mean that the mistake you made is a special kind of mistake: it is simply a mistake made intentionally. Similarly, suppose someone gives you a pen as a gift. It is perfectly true that the pen in question is a gift. But this does not mean that the pen is a special kind of object ( gift-pen as opposed to an ordinary pen). It is a gift simply because the pen was given in a certain way. Thus we maintain that Pierre is both an arbitrary Frenchman and a perfectly ordinary particular Frenchman. 2 Arbitrary reference and instantial reasoning The most obvious application of AR is to the interpretation of instantial reasoning. Much ordinary reasoning uses universal generalisation and existential instantiation. We would ideally like an account of such reasoning that satisfies two related constraints, a descriptive and a normative one. 12 On the descriptive side, we would like the account to correctly describe what it is that we are doing when making such inferences. More explicitly: when we make these inferences we use bits of language, and we would like an account of what those bits of language mean. On the normative side, we would like the account to explain why the inferences we make using UG and EI are justified. The two constraints are obviously linked: in so far as we believe that inferences made through instantial reasoning are justified, then an account would only be descriptively adequate if it also satisfies the second, normative constraint. In the discussion below it will be especially important to keep the descriptive constraint in mind: it is insufficient to suggest an account of instantial reasoning that provides a justification for some form of reasoning we could have been making but which in fact we do not make. Accounts which render instantial reasoning sound but are otherwise descriptively incorrect, are thus deficient. Having clarified what we expect from an account of instantial reasoning we will proceed to argue for two claims: that AR underlies a good account of instantial reasoning, and that this account fares better than the alternative accounts. If we are right, then by inference to the best explanation one has good reason to accept AR. It should be noted, though, that the issue of 12 Cf. Fine (1985b), p. 127, who suggests similar constraints. Page 7
how to interpret instantial reasoning is a complex one which has received a fair amount of discussion in the literature. 13 We cannot hope in the scope of this paper to discuss every alternative account in detail, and we do not take the section below to decisively refute every alternative account. But the discussion should at least give a sense of how difficult it is to provide a satisfactory account of instantial reasoning without appealing to AR. 2.1 Problems with alternative accounts Accounts of instantial reasoning fall into three kinds. First, there are instrumentalist accounts, according to which instantial terms 14 and the lines in the reasoning that include them are meaningless, and serve merely as convenient devices for reaching meaningful conclusions from meaningful premises. Second, there are quantificational accounts, according to which instantial terms are variables which are implicitly bound by quantifiers (or are themselves disguised quantifiers). Third, there are referential accounts, according to which instantial terms are names which refer to objects of some kind (arbitrary objects on Fine s account, ordinary objects on our own). We begin by presenting some of the main difficulties with the instrumentalist and quantificational accounts, drawing heavily on Fine s seminal discussion of the topic. 15 We then proceed to argue that Fine s own view also faces serious difficulties. It will be helpful throughout the discussion to have a particular argument involving instantial reasoning in mind. One paradigmatic argument which employs both UG and EI is the following argument, from the premise that there is someone who loves everyone to the conclusion that everyone is such that someone loves them. We present the argument, annotating each line with the rule of inference that seems, at least on the face of it, to be applied in each case: 16 Argument 1: 13 Literature on the topic includes at least Fine (1983), Fine (1985a), Fine (1985b), King (1991), Mackie (1985), Martino (2001), Price (1962), Rescher (1958), Shapiro (2004), and Tennant (1983). 14 An instantial term is term a, such that in an application of UG we infer xφ(x) from φ(a), or such that in an application of EI we infer φ(a) from xφ(x). 15 Fine (1985b), especially ch. 12. 16 The semi-formal English used in this argument is intended to help keep track of the relative scopes of the quantifiers. We assume that a full syntactic parsing of ordinary English will provide us with similar formal properties. Page 8
(1) There is someone x such that for every person y, x loves y [Premise] (2) Let John be such a person (3) For every person y, John loves y [Existential Instantiation on 1] (4) Let Jane be an arbitrary person (5) John loves Jane [Universal Instantiation on 3] (6) There is some person x such that x loves Jane [Existential Generalisation on 5] (7) But since Jane was an arbitrary person, for every person y there is some person x such that x loves y [Universal Generalisation on 6] 2.1.1 The instrumentalist account According to the instrumentalist account, instantial terms and thus the lines in which they occur are simply meaningless. The reason we nevertheless include such lines in our arguments is that they are instrumentally valuable: there is a purely syntactic or prooftheoretic theorem which ensures that if we manipulate these meaningless symbols in a specified set of ways then we will only infer true meaningful conclusions from true meaningful premises. The problem with this account is that it seems descriptively inadequate. According to the account, lines 2-6 in the above argument involve nothing but a meaningless manipulation of symbols. Yet it certainly seems to us that claims such as For every person y, John loves y in line 3 are not only meaningful, but (assuming we believe the premise) true. Moreover, according to the instrumentalist account, the only way to see that inferences involving instantial terms are valid is not by acknowledging that each step of the inference is individually acceptable, but rather by knowing some highly theoretical result in proof theory which shows that the relevant kinds of manipulations of symbols will ultimately yield correct results. But it at least seems that agents can perfectly well recognise the soundness of the above reasoning without knowing this general proof-theoretic result. The instrumentalist view thus seems a descriptively inadequate explanation of our practices. 2.1.2 Quantificational accounts Page 9
According to quantificational accounts, we should think of instantial terms as variables, ones that are implicitly bound by quantifiers. 17 The question, however, is how exactly are the instantial terms bound? At first, suppose that the proof as a whole occurs within the scope of some quantifiers: universal quantifiers binding instantial terms connected with applications of UG, and existential quantifiers binding instantial terms connected with applications of EI 18 (call this the wide scope theory ). While the wide scope theory might sound initially tempting, trying to reconstruct an actual argument following its recommendation immediately runs in difficulties. Consider argument 1 above. We could try to construe it as a long conjunction of the following sort (with j 1 and j 2 replacing John and Jane respectively): There is a person j 1 such that for every person y, j 1 loves y, and for all persons j 2 : There is someone x such that for every person y, x loves y & for every person y, j 1 loves y & j 1 loves j 2 & & for every person y, there is some person x such that x loves y. But this completely distorts the structure of the argument. First and most importantly, it seems that we are simply asserting each of the lines (including the conclusion!), rather than inferring it from previous lines. Second, on this construal some of the lines stand for open formulas, so are not individually truth-valued. Third, we may later extend our reasoning so as to include more lines which mention John and Jane. But then the new tokens of the instantial terms will fall outside of the scope of our quantifiers (or else we will have to assume that the new argument is not really an extension of the original one). And it is hard to see how different variants of the wide scope proposal will avoid similar problems. A much more promising proposal is to construe each step of the argument as individually bound by quantifiers (call this the narrow scope theory ). This has the advantage of allowing 17 King (1991) offers a quantificational account that treats the instantial terms themselves as implicit quantifiers, rather than as variables. This point will make no difference to our argument. 18 Many formal systems are phrased so that the same instantial term cannot act in both capacities. It is worth noting, though, that as long as we are careful to phrase the rules correctly, there should not be a principled difficulty in using a term for both purposes: an instantial term a used in an inference from xfx to Fa, can be treated as an arbitrary F, and generalised over to show that all Fs have a certain property. Page 10
each line of the argument to be truth-valued, and the argument as a whole to have a standard structure: it consists of a series of truth-valued statements, each of which follows from the previous steps. Still, construing instantial reasoning along the lines suggested by the narrow scope theory is far from smooth either. Consider for example the following argument, which from the premise that everything is F and everything is G, concludes that everything is F and G: Argument 2: (1) For all x, x is F [premise] (2) For all x, x is G [premise] (3) Let a be an arbitrary object (4) a is F [UI on 1] (5) a is G [UI on 2] (6) a is F and a is G [Conjunction Introduction on 4 and 5] (7) For all x, x is F and x is G [UG on 6] According to the narrow scope theory, lines 4, 5, and 6 say the same as lines 1, 2, and 7 respectively, so it is hard to see why we need these steps in the argument. And it is worth noting that one cannot reply to this worry by claiming that these lines serve the purpose of making explicit content that was previously expressed implicitly: after all, according to the narrow scope construal, lines 4 and 5 do precisely the opposite, namely restate content that was previously expressed explicitly in an implicit way. Another problem is that on the current construal step 6 is not actually obtained from steps 4 and 5 via conjunction introduction, as our provisional annotation suggests. Rather, it consists of a fairly complex inference from the claims that for all a, a is F and that for all a, a is G, to the claim that for all a, a is F and a is G. While this inference is sound, it is far from trivial indeed it is precisely the soundness of this inference that Argument 2 as a whole was supposed to establish! As Fine illustrates, things become even trickier when we consider certain applications of EI. Consider for example the following argument, from the premises that there are some French men and that everyone is tall, to the conclusion that there is a tall French man. Argument 3: (1) There are some French men [premise] Page 11
(2) Let Jack be one of them (3) Jack is a French man [EI on 1] (4) Everyone is tall [premise] (5) Jack is tall [UI on 4] (6) Jack is tall and Jack is a French man [Conjunction Introduction on 3 and 5] (7) So there is a tall French man [EG on 6] This seems like a perfectly good argument. But now consider how the narrow scope theory treats step 6. According to the theory, step 3 says (in a disguised way) that there is a French man, step 5 says (in a disguised way) that there is someone who is tall, and step 6 says (in a disguised way) that there is someone who is tall and a French man. This means that not only does step 6 not follow from 3 and 5 by Conjunction Introduction (as our annotation suggests), but in fact it does not follow from 3 and 5 at all: from the claim that there exists an F and there exists a G, it does not follow that there exists something that is both F and G. Thus according to the narrow scope theory, Argument 3 is simply illegitimate. 19 Finally, the narrow scope proposal has a problem accounting for the role of suppositions in arguments. Take an argument that contains the stipulation Let n be an arbitrary number. Further down the argument we might have the following line: Suppose that n is even, and we may then go on to show that certain things follow from this supposition. But, at least on its most straightforward version, the narrow scope theory construes the supposition as the claim that all numbers are even, which is clearly not what we intended. Having objected to the narrow scope theory in general, it is worth saying a few words on what is by far the most sophisticated version of the theory - the account of instantial reasoning presented in King (1991). One key advantage of King s account is that rather than suggesting the narrow scope theory in general terms, it offers a systematic way of determining (at least with respect to one particular formal system), precisely how each line in a proof should be interpreted. Another admirable feature of the account is that the interpretations are constructed in a careful way which avoids some of the technical worries mentioned above. Thus for example, argument 3 above is interpreted so that line 6 does 19 That is, unless, the argument is construed so that step 6 follows directly from steps 1 and 4, and step 5 is taken to be completely redundant. Page 12
follow from line 3 and 5. 20 And suppositions are treated so that each line within the scope of the supposition is in effect interpreted as a conditional, which has the supposition as its antecedent. Nevertheless, King s system does not manage to avoid the main problem facing the narrow scope theory: namely that it misconstrues the structure of arguments by instantial reasoning. Argument 2, for example, is construed in King s system just as we suggested above, where line 6 follows from lines 4 and 5 by a much more complex form of reasoning than Conjunction Introduction (indeed a form of reasoning the validity of which the whole argument was meant to establish). 21 Similarly, while the treatment of suppositions in King s system is technically adequate, it does not seem to correctly represent the structure of arguments involving suppositions. As Fine notes: In making the supposition φ I am not asserting the trivial conditional φ φ and in making an inference (say φ ψ) from a supposition (say φ) I am not inferring one conditional (φ ( φ ψ)) from another (φ φ). 22 Given King s admirable work of construing a precise system which predicts the specific list of quantifiers and their relative scopes in each line of the proof, we can now see an additional problem that the narrow scope theory faces: the theory is forced to make some highly arbitrary choices. Consider Argument 1. Roughly speaking, on King s system the statement John loves Jane in line 5 is interpreted with an existential quantifier within the scope of a universal quantifier ( There is a person x such that for every person y, x loves y ), and line 6 ( There is some person x such that x loves Jane ) is interpreted with a universal quantifier within the scope of an existential one ( For every person y there is some person x such that x loves y ). 23 This means that the crucial quantifier switch happens between lines 5 and 6 in the 20 Very roughly this is achieved by interpreting line 5 to say that some French man is tall, rather than merely that someone is tall. But this is actually a gross oversimplification. What King s system in fact predicts is the following interpretations for line 3, 5, 6 (with F standing for French man and T for tall): (3) y(( xfx Fy) Fy) y(( xfx Fy) Fy) (5) y(( xfx Fy) Ty) y(( xfx Fy) Ty) (6) y(( xfx Fy) Fy Ty) y(( xfx Fy) Fy Ty) The complexity of these interpretations should already give us serious cause for concern. 21 Things get even worse with other so called applications of Conjunction Introduction as we see in the argument from 3 and 5 to 6 in Argument 3 (see footnote 20). 22 Fine (1985b), p. 134. 23 We say roughly because again the system is more complex than that: line 5 will actually be interpreted as saying that a(( x ylxy ylay) blab) a(( x ylxy ylay) blab)), and line 6 as saying that a(( x ylxy ylay) b xlxb) a(( x ylxy ylay) b xlxb)). But what is crucial to the argument here are the claims that a blab and b xlxb which we get from the first conjuncts of the interpretations of Page 13
argument. But now suppose we move the stipulation that Jane be an arbitrary person (line 4), above line 2 (i.e. before applying EI to the premise). Now the argument will be construed on King s system so that line 5 ( John loves Jane ) has a different interpretation, one where the existential quantifier is in the scope of a universal quantifier ( For every person y, there is some person x, such that x loves y ), which means that the crucial quantifier switch already happened by the time we reach line 5. The upshot is that on King s account, it makes a crucial difference both to the interpretation of each line and to the structure of the argument more generally whether the stipulation Let Jane be an arbitrary person is made before or after the application of EI. But intuitively, there is no such difference between the two versions of the argument. By reflecting on King s construal of Arguments 1 through 3 we observe four problematic features of his account, which are characteristic of narrow scope accounts more generally. First, the account construes certain lines in the proof as mere repetitions of the premises or conclusion, sometimes rendering implicit, material that was already presented explicitly. Second, the account construes certain lines in the proof that seem to have a very simple structure (e.g. John loves Jane ) as making very complex claims ( a (( x ylxy ylay) blab) a(( x ylxy ylay) blab)) ). 24 Third, the account construes what seem to be very simple steps in reasoning, as actually employing quite complex forms of reasoning (e.g. the move from a scope structure to a scope structure) complex forms of reasoning the validity of which is often supposed to be established by the argument as a whole, rendering these steps in the argument question begging. Fourth, the account entails that seemingly arbitrary and unimportant choices in the placing of stipulations in the argument make a substantial difference to how the argument is interpreted. We conclude that quantificational accounts do not provide a satisfactory account of instantial reasoning. 2.1.3 Fine s referential account Instantial terms exhibit, at least on the face of it, the syntactic behaviour of proper names. This fact, coupled with the difficulties facing the quantificational accounts, provides a strong lines 5 and 6 respectively. We thus focus on these claims, and apply similar simplifications in the discussion below. 24 We are fully aware that King is in no way committed to saying that the original claim has the same syntactic structure as that of the complex interpretation. But even so, the semantic complexity of the interpretations is in itself worrisome. Page 14
reason to think that instantial terms are proper names, and serve to refer to objects. The most prominent view of instantial reasoning which takes instantial terms to be referential is Kit Fine s view. 25 We cannot hope in this space to do full justice to Fine s detailed and intricate discussion, but we would like to point out several aspects of Fine s view which are considerably problematic. According to Fine, instantial terms refer to special kinds of objects - arbitrary objects which are distinct from any particular object. Thus for example when we stipulate that n be an arbitrary number, we fix the reference of n to an arbitrary number, which is an entity distinct from any of the familiar particular numbers (2, 5467, and so forth). The rough picture is that each arbitrary object has a value-range associated with it which determines which properties it has: an arbitrary object will have all and only the properties that are shared by all the (particular) objects in its value-range. As Fine recognises, however, this rough picture requires substantial refinement. To begin with, we cannot accept without qualification that an arbitrary object has all and only the properties shared by all the objects in its value-range: the arbitrary number is an arbitrary object, even though not all of the objects in its value-range are arbitrary objects (in fact none of them are). Similarly, the arbitrary person is an abstract object, even though not all of the objects in its value-range are abstract objects (in fact none of them are). In order to address this problem, Fine distinguishes between what he calls generic and classical conditions. Generic conditions are properties such as being a number or being even, ones that hold of an arbitrary object if and only if they hold of every object in its value-range. Classical conditions, on the other hand, are properties such as being arbitrary ones which do not satisfy this principle. The distinction between generic and classical conditions looks suspiciously like an ad-hoc fix. But even if we grant Fine this distinction, there are further problems. Consider the following question: how many natural numbers are there between one and ten? Intuitively, the answer is ten. But it seems that Fine s theory predicts otherwise: according to Fine, in addition to all the particular numbers between one and ten, there is also the arbitrary number 25 See Fine (1983), Fine (1985a), and Fine (1985b) Page 15
between one and ten (call it Arb ). So there must be at least eleven numbers between one and ten! Does the distinction between classical and generic conditions help with this problem? It is not clear that it does. After all, being a number between one and ten seems like a paradigmatic example of a generic condition; if so, then since each of the objects in the value-range of Arb satisfies this condition, Arb satisfies it as well so Arb is a number between one and ten. A further attempt to respond might appeal to Fine s suggestion that there may be cases in which a predicate is ambiguous as between a generic and classical reading. The predicate is a number is a good example. On a generic reading, it is inclusive of all arbitrary numbers; on a classical reading it is exclusive of them. 26 The proposal is that on a classical reading of number between one and ten, Arb is not a number between one and ten, while on a generic reading it is. So the statement There are exactly ten numbers between one and ten is true on one reading and false on the other. We find this response unsatisfactory. For a start, it seems that the sentence There are exactly ten numbers between one and ten has only a true reading, and no false reading. Moreover, we see no reason to believe that number between one and ten is ambiguous in the manner Fine suggests. 27 Consider for example the following speech: There are ten numbers between one and ten. Let Arb be (an arbitrary) one of them. This seems like a perfectly standard way of introducing an instantial term, but Fine s ambiguity proposal predicts otherwise. Since the term them is anaphoric upon the term numbers between one and ten in the preceding sentence, both phrases must receive the same interpretation. So either number between one and ten is interpreted classically, in which case Arb cannot be an arbitrary number which has this property, or number between one and ten is interpreted generically, in which case the initial claim that there are ten numbers between one and ten is false. Next we turn to a second refinement which Fine s theory requires, one which concerns the relationship between distinct instantial terms appearing in the same argument. Consider the stipulation Let m be an arbitrary real number and let k be an arbitrary real number greater than m. According to Fine s theory, m and k will both be arbitrary real numbers, and will 26 Fine (1985b), p. 14. 27 Note also that since the counting problem seems to generalize, Fine would need to argue that pretty much every predicate is ambiguous in this manner. Page 16
have as their value-range the full set of real numbers. But this cannot be the whole story: we want somehow to capture the idea that k and m are related, and in particular that k must be greater than m. To that end, Fine proposes two additional pieces of machinery. The first is the idea that a collection of arbitrary objects may have joint rather than individual valueranges. So, for example, instead of saying that m and k each have the full range of real numbers as their value-range, we can say that m, k jointly have as their value-ranges pairs of real numbers (pairs where the second member is larger than the first member). The second piece of machinery involves the idea that arbitrary objects come in two kinds: dependent and independent. While m is an independent arbitrary number, k is a dependent arbitrary number: in a sense its value depends on the value of m. Other than adding a further layer of complication to the theory, this complex hierarchy of arbitrary objects runs into trouble when we consider the relation of identity between them. Consider the stipulation Let m be an arbitrary number, and let k be an arbitrary number. One natural suggestion is to assume that m and k are both independent arbitrary objects, which have all numbers as their value-range. The problem is that Fine insists that there is only one independent arbitrary object associated with each value-range. 28 But this in turn means that we would be able to infer that m=k, which we clearly ought not to. One could try responding to this problem by appealing to the ambiguity between classical and generic readings, claiming that = is ambiguous in this manner. But this will not do: first, it is not clear how we get a generic reading of = here: after all, as applied to independent arbitrary object the generic reading is supposed to be interpreted by considering the range of values for each of the objects independently. Second, in so far as we can make sense of the generic reading for =, it will be one according to which it is not true that m=k (because it is not the case that all potential values for m and k will be identical to each other). But just as we ought not to infer from the above stipulation that m=k, equally we ought not to infer that m k! Third, as Fine defined the classical reading of a predicate, it is one where as applied to an arbitrary object it would be automatically false. So it turns out that both the classical reading and the generic one should yield the result that m k, and thus we cannot appeal to the alleged ambiguity of = to justify our indecision as to whether m=k or m k. Finally, we note that the ambiguity theory is particularly unappealing in the case of identity: perhaps more 28 See Fine (1985b), p. 18. Fine later relaxes this condition (ibid. p. 35), but claims he does this only because for certain technical purposes, a smoother theory is obtained. Page 17
than any other predicate, we have a clear sense that identity has a complete and determinate definition, one that already applies in the standard way to any kind of object (even to arbitrary ones, if there are such things). A different response to the identity problem would be to suggest that there can be more than one independent arbitrary object with the same value-range. So perhaps following the above stipulation, it is possible that m and k refer to the same arbitrary number but also possible that they refer to two qualitatively identical but numerically distinct arbitrary numbers, thus accounting for our indecision. The problem is that if this proposal is adopted, one would need to explain how it is that the reference of m is fixed to one independent arbitrary number rather than another. And given that the different independent arbitrary numbers are qualitatively identical, it is hard to see how could give such story, without already appealing to something like AR. 29 But as we will go on to show, if one already accepts AR, there is a much simpler theory of instantial reasoning, and thus there is no need for Fine s theory of arbitrary objects in the first place. A final solution to the identity problem is to suggest that both m and k are dependent rather than independent arbitrary objects: perhaps they both depend upon the arbitrary pair of numbers, p. The value range of p is the set of pairs of numbers; m depends upon p in this manner: the value of m must be the same as the first element of the value of p; k depends upon p in this different manner: the value of k must be the same as the second element of the value of p. 30 But this solution will not do either. For a start the solution requires that the referent of m is fixed to one object (the independent arbitrary number) if the stipulation Let m be an arbitrary number appears on its own in an argument, but fixed to an entirely different object (one that is dependent upon the arbitrary pair of numbers in the manner described above) if the stipulation Let n be an arbitrary number appears later in the argument. But this seems entirely wrong: the two stipulations are completely unrelated, and it is odd that we cannot fix the reference of m until we see what other instantial terms appear further down the argument. As Fine himself notes (in another context): It is a natural requirement on a derivation containing A-names that we know what those names denote as soon as they are introduced; their interpretation should not depend upon what subsequently 29 Cf. 3.2 below. 30 This seems to be the solution favoured by Fine (see Fine (1985b), p. 19). Page 18
happens in the derivation. 31 An even graver problem with the current proposed solution is that it too would lead to the undesirable conclusion that m k: two dependent arbitrary objects are identical, according to the theory, if and only if they depend on the same arbitrary objects in exactly the same manner. But m depends on the arbitrary ordered pair in a different manner than k does, so we should be able to infer that the two are not identical, i.e. that m k. Leaving the identity problem, we go on to note a third layer of complication for Fine s theory. This has to do with the observation that Fine s theory seems to face the same objection concerning the role of suppositions that the narrow-scope quantificational view faced. Consider a proof containing the stipulation Let n be an arbitrary number, where later we add a supposition such as Suppose n is even. Since on Fine s view an arbitrary number n is even if and only if every number is even, this suggests that the content of the supposition is that every number is even, which is clearly not what we want to suppose. Fine s response to this problem involves introducing the notion of a vacancy value : 32 Arbitrary objects are split into vacant and occupied objects, where an occupied object is one that is treated (in some sense) as if it has been assigned one particular value, while a vacant arbitrary object is treated in full generality. The logic is then amended so that vacancy values are marked, and inferences involving suppositions as above can only be made where the supposition is interpreted as involving an occupied arbitrary object. We will not discuss the details of this proposal, but we merely note that it adds yet another complication to Fine s theory, one that involves a technical apparatus which is hard to motivate on independent grounds, or to interpret in any natural way. Our final major worry concerning Fine s theory is that it involves a rejection of classical logic. One way to see this is to note that according to Fine, the basic principle is that a sentence concerning A-objects is true (false) just in case it is true (false) for all of their values. 33 This in turn entails that it is neither true nor false that the arbitrary number is even, and the principle of bivalence must be rejected. A related issue concerns the semantics of the connectives: it is true that the arbitrary number is either odd or even, even though it is not true that it is odd and it is not true that it is even. Fine is happy to endorse the rejection of classical logic (he notes that the phenomenon of vagueness should anyhow motivate us 31 Fine (1985b), p. 101. 32 Fine (1985b), p. 75-80. 33 Fine (1985b), p. 41. Page 19